1. Number Systems & Operations
Part I

2. Decimal Numbers (Base 10)
People use decimal numbers.
I hope you know this very well. However, let’s review:
Ten digits  0-9
The value of a digit is determined by its position in the number.

3. Binary Numbers
Decimal
Binary 1 2 10 3 11 4
100 5
101 6
110 7
111 8
1000 9
1001
1010
There are only 2 digits (0 and 1) and we can do binary counting as shown in the table.

4. Binary Numbers (Base 2) The weighting structure of binary numbers
2n n
Positive power of two
(whole number)
Negative power of two
(fractional number) 25 24 23 22 21 20
2-1
2-2
2-3
2-4
2-5
2-6 32 16 8 4 2 1
1/2
1/4
1/8
1/16
1/32
1/64
0.5
0.25
0.125
0.0625

5. Binary-to-Decimal Conversion
Add the weights of all 1s in a binary number to get the decimal value.
ex: convert to decimal = =
= 109
Weight 26 25 24 23 22 21 20
bin 1

6. Binary-to-Decimal Conversion
Fractional binary example
ex: convert to decimal = = =
Weight
2-1
2-2
2-3 24
bin 1

7. Decimal-to-Binary Conversion
Sum-of-weights method
To get the binary number for a given decimal number, find the binary weights that add up to the decimal number.
ex: convert 1210 , 2510 , 5810 , 8210 to binary
12 = 8+4 = = 1100
25 = = = 11001
58 = = =
82 = = =

8. Decimal-to-Binary Conversion
Repeated division-by-2 method
To get the binary number for a given decimal number, divide the decimal number by 2 until the quotient is 0. Remainders form the binary number.
remainder
12/2 = 6
6/2 3
3/2 1
1/2
LSB
MSB
Stop when the whole-number quotient is 0
1210 = 11002

9. Decimal-to-Binary Conversion
Converting decimal fractions to binary
Sum-of-weights
This method can be applied to fractional decimal numbers, as shown in the following example:
0.625 = = = 0.101
Repeated multiplication by 2
Decimal fraction can be converted to binary by repeated multiplication by 2 (see details in the following slide.)

10. Repeated Multiplication by 2 (by example)
ex: convert the decimal fraction to binary
carry
x 2 =
0.625
0.625 x 2
1.25 1
0.25 x 2
0.50
0.50 x 2
1.00
MSB
LSB
Continue to the desired number of decimal places or stop when the fractional part is all zero =

11. Binary Arithmetic Basic of binary arithmetic Binary addition
Binary subtraction
Binary multiplication
Binary division

12. Binary Addition The four basic rules for adding digits are as follows:
0+0=0  sum of 0 with a carry of 0
0+1=1  sum of 1 with a carry of 0
1+0=1  sum of 1 with a carry of 0
1+1=10  sum of 0 with a carry of 1

13. Binary Addition (by example)
11 3

14. Binary Subtraction
The four basic rules for subtracting digits are as follows:
0-0 = 0
1-1 = 0
1-0 = 1
10-1 = 1 ; 0-1 with a borrow of 1

15. Binary Subtraction (by example)
11 3
10 2
11 3
01 1

16. Binary Multiplication
The four basic rules for multiplying digits are as follows:
0x0 = 0
0x1 = 0
1x0 = 0
1x1 = 1
Multiplication is performed with binary numbers in the same manner as with decimal numbers.
It involves forming partial products, shifting each successive partial product left one place, and then adding all the partial products.

17. Binary Multiplication (by example)11
x11
+11
1001 3 x3 9
101
x111
+101
100011 5 x7 35

18. Binary Division
Division in binary follows the same procedure as division in decimal. 11
10 110 10 00 3
2 6 6 10
11 110 11
000 2
3 6 6

19. 1’s and 2’s Complements
They are important since they permit the presentation of negative numbers.
The method of 2’s complement arithmetic is commonly used in computers to handle negative numbers.

20. Finding the 1’s complement
Very simple: change each bit in a number to get the 1’s complement
ex: find 1’s complement of
Binary 1
1’s complement

21. Finding the 2’s Complement
Add 1 to the 1’s complement to get the 2’s complement.
ex:  
An alternative method:
Start at the right with the LSB and write the bits as they are up to and including the first 1.
Take the 1’s complement of the remaining bits.
binary
’s comp +1
1’s complement
2’s complement

22. Signed Numbers
Digital systems, such as computer, must be able to handle both positive and negative numbers.
A signed binary number consists of both sign and magnitude information.
The sign indicates whether a number is positive or negative.
The magnitude is the value of the number.

23. Signed Numbers
There are 3 forms in which signed integer numbers can be represented in binary:
Sign-magnitude (least used)
1’s complement
2’s complement (most important)
Non-integer and very large or small numbers can be expressed in floating-point format.

24. The Sign Bit
The left-most bit in a signed binary number is the sign bit.
It tells you whether the number is positive (sign bit = 0) or negative (sign bit = 1).

25. Sign-Magnitude Form
The left-most bit is the sign bit and the remaining bits are the magnitude bits.
The magnitude bits are in true binary for both positive and negative numbers.
ex: the decimal number +25 is expressed as an 8-bit signed binary number as:
While the decimal number -25 is expressed as

26. Sign-Magnitude Form
“ In the sign-magnitude form, a negative number has the same magnitude bits as the corresponding positive number but the sign bit is a 1 rather than a 0. “

27. 1’s Complement Form
Positive numbers in 1’s complement form are represented the same way as the positive sign-magnitude.
Negative numbers are the 1’s complements of the corresponding positive numbers.
ex: the decimal number +25 is expressed as:
While the decimal number -25 is expressed as

28. 1’s Complement Form
“ In the 1’s complement form, a negative number is the 1’s complement of the corresponding positive number. “

29. 2’s Complement Form
Positive numbers in 2’s complement form are represented the same way as the positive sign-magnitude and 1’s complement form.
Negative numbers are the 2’s complements of the corresponding positive numbers.
ex: the decimal number +25 is expressed as:
While the decimal number -25 is expressed as

30. 2’s Complement Form
“ In the 2’s complement form, a negative number is the 2’s complement of the corresponding positive number. “

31. Decimal Value of Signed Numbers
Sign-magnitude:
Both positive and negative numbers are determined by summing the weights in all the magnitude bit positions where these are 1s and ignoring those positions where there are 0s.
The sign is determined by examination of the sign bit.

32. Decimal Value of Signed Numbers
Sign-magnitude (by example)
ex: decimal values of these numbers (expressed in sign-magnitude)
1) )
1) )
magnitude magnitude
= = = = 119
sign sign
= 1  negative = 0  positive
Hence: = Hence: = 119

33. Decimal Value of Signed Numbers
1’s complement:
Positive – determined by summing the weights in all bit positions where there are 1s and ignoring those positions where there are 0s.
Negative – determined by assigning a negative value to the weight of the sign bit, summing all the weights where there are 1’s, and adding 1 to the result.

34. Decimal Value of Signed Numbers
1’s complement (by example)
ex: decimal values of these numbers (expressed in 1’s complement)
1) )
1) )
= = = (-128) = -24
Hence: = Hence: = -23 +1

35. Decimal Value of Signed Numbers
2’s complement:
Positive – determined by summing the weights in all bit positions where there are 1s and ignoring those positions where there are 0s.
Negative – the weight of the sign bit in a negative number is given a negative value.

36. Decimal Value of Signed Numbers
2’s complement (by example)
ex: decimal values of these numbers (expressed in 2’s complement)
1) )
1) )
= = = (-128) = -86
Hence: = Hence: = -86

37. Range of Signed Integer Numbers
The range of magnitude of a binary number depends on the number of bits (n).
Total combinations = 2n
8 bits = 256 different numbers
16 bits = 65,536 different numbers
32 bits = 4,294,967,296 different numbers

38. Range of Signed Integer Numbers
For 2’s complement signed numbers:
Range = -(2n-1) to +(2n-1-1)
where there is one sign bit and n-1 magnitude
ex:
Negative Boundary
Positive Boundary
4 bits
-(23) = -8
(23-1) = +7
8 bits
-(27) = -128
(27-1) = +127
16 bits
-(215) = -32,768
(215-1) =

39. Floating-point numbers
How many bits do we need to represent very large number?
Floating-point number consists of two parts plus a sign.
Mantissa – represents the magnitude of the number.
Exponent – represents the number of places that the decimal point (or binary point) is to be moved.
Decimal number example: 241,506,800
Mantissa =
Exponent = 109
Can be written as FP as x 109

40. Binary FP Numbers The format defined by ANSI/IEEE Standard 754-1985
Single-precision
Double-precision
Extended-precision
Same basic formats except for the number of bits.
Single-precision = 32 bits
Double-precision = 64 bits
(Double) Extended-precision = 80 bits

41. Single-Precision Floating-Point Binary Numbers
Standard format:
Sign bit (S) – 1 bit
Exponent (E) – 8 bits
Mantissa or fraction (F) – 23 bits
S(1)
E(8)
F(23)
Single-precision FP Binary Number Format

42. Single-Precision Floating-Point Binary Numbers
Mantissa
The binary point is understood to be to the left of the 23 bits.
Effectively, there are 24 bits in the mantissa because in any binary number the left most bit is always 1. (say is )
Therefore, this 1 is understood to be there although it does not occupy an actual bit position.
S(1)
E(8)
F(23)
Single-precision FP Binary Number Format

43. Single-Precision Floating-Point Binary Numbers
Exponent
The eight bits represent a biased exponent which is obtained by adding 127.
The purpose of the bias is to allow very large or very small numbers without requiring a separate sign bit for the exponents.
The biased exp allows a range of actual exp values from -126 ( ) to +127 ( )
S(1)
E(8)
F(23)
Single-precision FP Binary Number Format

44. Single-Precision Floating-Point Binary Numbers
Not easy, is it? Let’s see an example.
ex: (assumption: positive number)
It can be expressed as 1 plus a fractional binary number.
Hence:
= x 212
The exponent,12, is expressed as a biased exponent as followed:
= 139 =
Therefore, we get:

45. Single-Precision Floating-Point Binary Numbers
Let’s do the opposite way:
To evaluate a binary number in FP format.
General formula:
Number = (-1)S(1+F)(2E-127)
ex:
Number = (-1)( )( )
= (-1)( )(218) =
= -407,68010 1

46. Single-Precision Floating-Point Binary Numbers
Let’s review:
The exponent can be any number between to +128; that means extremely large and small numbers can be expressed.
Say, a 32-bit FP number can replace a binary integer number having 129 bits.
Distinctive point: Because the exponent determines the position of the binary point, numbers containing both integer and fractional parts can be represented.

47. Single-Precision Floating-Point Binary Numbers
There are 2 exceptions to the format for FP numbers:
The number 0.0 is represented by all 0s.
Infinity is represented by all 1s in the exponent and all 0s in the mantissa. x x

48. Let’s Practice…
Convert the decimal number 3.248x104 to a single-precision floating-point binary number